Identify the function (f(z)) and the point (z_0)
Check whether (z_0) is an isolated singularity
If (z_0) is a simple pole, use (operatorname{Res}(f,z_0)=lim_{zto z_0}(z-z_0)f(z))
If (f(z)=dfrac{g(z)}{h(z)}) and (h(z_0)=0) with (h'(z_0)neq 0), use (operatorname{Res}(f,z_0)=dfrac{g(z_0)}{h'(z_0)})
If (z_0) is a pole of order (m), use (operatorname{Res}(f,z_0)=dfrac{1}{(m-1)!}lim_{zto z_0}dfrac{d^{,m-1}}{dz^{m-1}}left[(z-z_0)^m f(z)right])
If (f) has a Laurent series around (z_0), take the coefficient of ((z-z_0)^{-1})
If (f(z)=sum a_n(z-z_0)^n) is known in Laurent form, the residue is (a_{-1})
For rational functions, factor the denominator and use partial fractions
For products, expand the factor that is analytic at (z_0) into a Taylor series
For essential singularities, compute the Laurent expansion and extract the ((z-z_0)^{-1}) term
For residues at infinity, use (operatorname{Res}(f,infty)=-sum operatorname{Res}(f,z_k)) over all finite poles
Verify the result by checking the local expansion near (z_0)